Solve for x 2^(2x)-32*2^x+256=0

Problem

2^(2*x)−32⋅2^x+256=0

Solution

1. Substitute a new variable to simplify the equation by letting u=2^x

2. Rewrite the equation in terms of u noting that 2^(2*x)=(2^x)^2=u^2

u^2−32*u+256=0

3. Factor the quadratic equation by recognizing it as a perfect square trinomial, since 256=16^2 and 32=2⋅16

(u−16)^2=0

4. Solve for u by taking the square root of both sides.

u−16=0

u=16

5. Back-substitute the original expression 2^x for u

2^x=16

6. Solve for x by expressing 16 as a power of 2.

2^x=2^4

x=4

Final Answer

x=4

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